Euler’s phi function 2,8,48, 480, 5760


It doesn’t take too much delving around with the primes to come across the appearance of Euler’s totient or phi function. It oozes out of the primes in a variety of locations.

Firstly the composite helices and thus the prime helices each have a “width”. The width of the 6n+/-1 helix associated with prime factor 5 is 2. The width of the next helix associated with prime factor 7 is 8. The next is 48 and the conjecture is that this continues:

480

5760

etc

Secondly, considering the balance of primes to composites as each prime factor is used to calculate which are composite and which are potentially prime we see a simple pattern:

Each block is pi# integers long,

Each block has (pi   – 1) # (potential) primes &  (pi-1 -1)# composites

Thus the ratio of

primes to all integers:    (pi   – 1) # /  pi#

primes to composites       (pi   – 1) # / (pi-1 -1) # = pi   – 1

 

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